Higher structures in Differential Geometry and Homotopy Theory

PI1: Antonio Viruel
PI2: Aniceto Murillo

Abstract: The aim of this project is to study higher structures in both Differential Geometry and Homotopy Theory. So we plan to attain the following goals:
(O1): On L-infinite algebras: We give a completely new approach to rational homotopy theory from the Quillen framework which extends his theory and will let us attack deep problems not only in homotopy theory,  but  from deformation to number theory, from a topological point of view.
(O2): On Sullivan models and self homotopy equivalences: We realize algebraic linear groups defined over Q as groups of self homotopy equivalences of a rational space. The techniques developed will allow us to codify combinatorial problems by means of Sullivan models. 
(O3): On classifying spaces: We generalize classical results on p-local group cohomology and homotopy types within the framework of fusion systems. 
(O4): On homotopy invariants: We calculate LS type invariants for Lie groups and homogeneous non symmetric spaces. We plan to provide Ganea and Whitehead approaches to simplicial LS category.
(O5): Diffeomorphisms groups: We find both diffeomorphisms groups and describe vector fields that are compatible with higher geometrical structures. We shall work on the existence of Jordan bounds on diffeomorphisms groups and the rank of a manifold defined in terms of vector fields with null brackets.
(O6): Semi-riemannian geometry: We deep in the rigging technique. We expect to obtain new bounds on the injectivity radius of light cones. We plan to construct rigged metrics on light hypersurfaces  endowed with special properties, e.g. Sasaki metrics. We also aim to provide new perspectives on the mathematical formulation of some  concepts of Relativity.